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Mathematische Zeitschrift

, Volume 284, Issue 1–2, pp 23–39 | Cite as

On complex zeros off the critical line for non-monomial polynomial of zeta-functions

  • Takashi Nakamura
  • Łukasz Pańkowski
Article

Abstract

In this paper, we show that any polynomial of zeta or L-functions with some conditions has infinitely many complex zeros off the critical line. This general result has abundant applications. By using the main result, we prove that the zeta-functions associated to symmetric matrices treated by Ibukiyama and Saito, certain spectral zeta-functions and the Euler–Zagier multiple zeta-functions have infinitely many complex zeros off the critical line. Moreover, we show that the Lindelöf hypothesis for the Riemann zeta-function is equivalent to the Lindelöf hypothesis for zeta-functions mentioned above despite of the existence of the zeros off the critical line. Next we prove that the Barnes multiple zeta-functions associated to rational or transcendental parameters have infinitely many zeros off the critical line. By using this fact, we show that the Shintani multiple zeta-functions have infinitely many complex zeros under some conditions. As corollaries, we show that the Mordell multiple zeta-functions, the Euler–Zagier–Hurwitz type of multiple zeta-functions and the Witten multiple zeta-functions have infinitely many complex zeros off the critical line.

Keywords

Hybrid universality Lindelöf hypothesis Zeros of zeta-functions associated to symmetric matrices Euler–Zagier multiple zeta-functions Spectral zeta-functions Barnes multiple zeta-functions Shintani multiple zeta-functions 

Mathematics Subject Classification

Primary 11M26 11M32 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Liberal Arts, Faculty of Science and TechnologyTokyo University of Science NodaChibaJapan
  2. 2.Faculty of Mathematics and Computer ScienceAdam Mickiewicz UniversityPoznanPoland

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